Modeling and Pricing Longevity Derivatives Using Stochastic Mortality Rates and the Esscher Transform

The Lee-Carter mortality model provides a structure for stochastically modeling mortality rates incorporating both time (year) and age mortality dynamics. Their model is constructed by modeling the mortality rate as a function of both an age and a year effect. Recently the MBMM model (Mitchell et al. 2013) showed the Lee Carter model can be improved by fitting with the growth rates of mortality rates over time and age rather than the mortality rates themselves. The MBMM modification of the Lee-Carter model performs better than the original and many of the subsequent variants. In order to model the mortality rate under the martingale measure and to apply it for pricing the longevity derivatives, we adapt the MBMM structure and introduce a Lévy stochastic process with a normal inverse Gaussian (NIG) distribution in our model. The model has two advantages in addition to better fit: first, it can mimic the jumps in the mortality rates since the NIG distribution is fat-tailed with high kurtosis, and, second, this mortality model lends itself to pricing of longevity derivatives based on the assumed mortality model. Using the Esscher transformation we show how to find a related martingale measure, allowing martingale pricing for mortality/longevity risk–related derivatives. Finally, we apply our model to pricing a q-forward longevity derivative utilizing the structure proposed by Life and Longevity Markets Association.     

A Two-Factor Model for Stochastic Mortality with Parameter Uncertainty: Theory and Calibration

In this article, we consider the evolution of the post‐age‐60 mortality curve in the United Kingdom and its impact on the pricing of the risk associated with aggregate mortality improvements over time: so‐called longevity risk. We introduce a two‐factor stochastic model for the development of this curve through time. The first factor affects mortality‐rate dynamics at all ages in the same way, whereas the second factor affects mortality‐rate dynamics at higher ages much more than at lower ages. The article then examines the pricing of longevity bonds with different terms to maturity referenced to different cohorts. We find that longevity risk over relatively short time horizons is very low, but at horizons in excess of ten years it begins to pick up very rapidly. A key component of the article is the proposal and development of a method for calculating the market risk‐adjusted price of a longevity bond. The proposed adjustment includes not just an allowance for the underlying stochastic mortality, but also makes an allowance for parameter risk. We utilize the pricing information contained in the November 2004 European Investment Bank longevity bond to make inferences about the likely market prices of the risks in the model. Based on these, we investigate how future issues might be priced to ensure an absence of arbitrage between bonds with different characteristics.     

The Statistical and Economic Role of Jumps in Continuous-Time Interest Rate Models

This paper analyzes the role of jumps in continuous‐time short rate models. I first develop a test to detect jump‐induced misspecification and, using Treasury bill rates, find evidence for the presence of jumps. Second, I specify and estimate a nonparametric jump‐diffusion model. Results indicate that jumps play an important statistical role. Estimates of jump times and sizes indicate that unexpected news about the macroeconomy generates the jumps. Finally, I investigate the pricing implications of jumps. Jumps generally have a minor impact on yields, but they are important for pricing interest rate options.     

EXACT FORMULAS FOR PRICING BONDS AND OPTIONS WHEN INTEREST RATE DIFFUSIONS CONTAIN JUMPS

I develop Heath‐Jarrow‐Morton extensions of the Vasicek and Jamshidian pure‐diffusion models, extend these models to incorporate Poisson‐Gaussian interest rate jumps, and obtain closed‐form models for valuing default‐free, zero‐coupon bonds and European call and put options on default‐free, zero‐coupon bonds in a market where interest rates can experience discontinuous information shocks. The jump‐diffusion pricing models value the instrument as the probability‐weighted average of the pure‐diffusion model prices, each conditional on a specific number of jumps occurring during the life of the instrument. I extend the models to coupon‐bearing instruments by applying Jamshidian's serial‐decomposition technique.     

Canonical Valuation of Mortality‐Linked Securities

A fundamental question in the study of mortality‐linked securities is how to place a value on them. This is still an open question, partly because there is a lack of liquidly traded longevity indexes or securities from which we can infer the market price of risk. This article develops a framework for pricing mortality‐linked securities on the basis of canonical valuation. This framework is largely nonparametric, helping us avoid parameter and model risk, which may be significant in other pricing methods. The framework is then applied to a mortality‐linked security, and the results are compared against those derived from other methods.     

长寿债券的运行机制与定价模型

通过分析长寿债券的市场发展以及连续型和触发型两类长寿债券的运行机制,采用风险中性定价方法推导出当死亡率服从双指数跳跃(DEJD)分布时,长寿债券的定价解析式,研究发现,无论从理论还是实践看,设计并发行触发型长寿债券是一种应对长寿风险更为明智的选择。     

Expected returns, risk premia, and volatility surfaces implicit in option market prices

This article presents a pure exchange economy that extends Rubinstein (1976) to show how the jump-diffusion option pricing model of Merton (1976) is altered when jumps are correlated with diffusive risks. A non-zero correlation between jumps and diffusive risks is necessary in order to resolve the positively sloped implied volatility term structure inherent in traditional jump diffusion models. Our evidence is consistent with a negative covariance, producing a non-monotonic term structure. For the proposed market structure, we present a closed form asset pricing model that depends on the factors of the traditional jump-diffusion models, and on both the covariance of the diffusive pricing kernel with price jumps and the covariance of the jumps of the pricing kernel with the diffusive price. We present statistical evidence that these covariances are positive. For our model the expected stock return, jump and diffusive risk premiums are non-linear functions of time.     

Mortality regimes and longevity risk in a life annuity portfolio

This paper explores the presence of changes of trends or jumps in French mortality from 1947 to 2007, and assesses their implications on the longevity risk management of a life annuity portfolio. We accomplish this by extending the Poisson log-bilinear regression developed by Brouhns et al. (2002) with a regime-switching model. Estimation results show that French mortality is characterized by two distinct regimes. One refers to a strong uncertainty state, which corresponds to the longevity conditions observed during the decade following World War II. The second regime is related to the low volatility of longevity improvements observed during the last 30 years. We use these results to analyze the impact of mortality regimes on the longevity risk management of a life annuity portfolio. Simulation results suggest that the changes of trends in the mortality process have some implications for longevity risk management.     

动态死亡率下个人年金的长寿风险分析

传统的精算定价方法假定死亡率是静态的,实际上死亡率是随时间而变动的具有动态不确定性的变量。在动态死亡率的框架下定量分析长寿风险对于个人年金产品定价的影响:引入Wang转换的风险定价方法度量长寿风险的市场价格,并运用模拟分析的方法分析长寿风险对个人年金定价的影响。最后,基于分析结果,就保险公司如何管理这一风险给出建议。     

The Political Economy of Government-Issued Longevity Bonds

This article explores the trade‐offs associated with government issuance of longevity bonds as a way of stimulating private annuity supply in the presence of aggregate mortality risk. We provide new calculations suggesting a 5 percent chance that aggregate mortality risk could ex post raise annuity costs for private insurers by as much as 5–10 percentage points, with the most likely effect based on historical patterns toward the lower end of that range. While we suspect that aggregate mortality risk does exert some upward pressure on annuity prices, evidence from private market pricing suggests that, to the extent that private insurers are accurately pricing this risk, the effect is less than 5 percentage points. We discuss ways that the private market can spread this risk, while emphasizing that the government has the unique ability to spread aggregate risk across generations. We note factors that might hamper such an efficient allocation of risk, including potential political incentives for the government to shift more than the optimal amount of risk onto future generations, and the possibility that government fiscal policy might allocate risk less efficiently within each generation than would private markets. We also discuss how large‐scale longevity bond issuance might affect government borrowing costs, as well as political economy aspects of how the proceeds from such a bond issuance might be used.     

Pricing derivatives with modeling CO2 emission allowance using a regime-switching jump diffusion model: with regime-switching risk premium

Carbon markets trade the spot European Union Allowance (EUA), with one EUA providing the right to emit one tone of carbon dioxide (CO₂). We examine the spot EUA returns in BlueNext that exhibit jumps and a volatility clustering feature. We propose a regime-switching jump diffusion model (RSJM) with a hidden Markov chain to capture not only a volatility clustering feature, but also the dynamics of the spot EUA returns that are influenced by change in the CO₂ emission economic conditions. In addition, the switching jump intensities of the RSJM are shown to be affected by change in the carbon-market macroeconomic environment. We further derive the theoretical futures-option prices with a constant convenience yield under the RSJM via the generalized Esscher transform where regime-switching risk is priced with a risk premium. The empirical study shows that the derived futures-option pricing model under the RSJM with regime-switching risk is a more complete model than a jump diffusion model for pricing CO₂ options.     

Pricing American interest rate options under the jump-extended constant-elasticity-of-variance short rate models

This paper demonstrates how to value American interest rate options under the jump-extended constant-elasticity-of-variance (CEV) models. We consider both exponential jumps (see Duffie et al., 2000) and lognormal jumps (see Johannes, 2004) in the short rate process. We show how to superimpose recombining multinomial jump trees on the diffusion trees, creating mixed jump-diffusion trees for the CEV models of short rate extended with exponential and lognormal jumps. Our simulations for the special case of jump-extended Cox, Ingersoll, and Ross (CIR) square root model show a significant computational advantage over the Longstaff and Schwartz’s (2001) least-squares regression method (LSM) for pricing American options on zero-coupon bonds.     

A General Semi-Markov Model for Coupled Lifetimes

Joint-life annuities with a high last survivor benefit play an important role in the optimal annuity portfolio for a retired couple. The dependence between coupled lifetimes is crucial for valuing joint-life annuities. Existing bivariate modeling of coupled lifetimes is based on outdated data with limited observation periods and does not take into account mortality improvement. In this article, we propose a transparent and dynamic framework for modeling coupled lifetime dependence caused by both marital status and common mortality improvement factors. Dependence due to marital status is captured by a semi-Markov joint life model. Dependence due to common mortality improvement, which represents the correlation between mortality improvement patterns of coupled lives, is incorporated by a two-population mortality improvement model. The proposed model is applied to pricing the longevity risk in last survivor annuities sold in the United States and the United Kingdom.     

基于双因素Wang转换方法的长寿风险债券定价研究

在比较国外经典债券设计的基础上,基于离散型死亡率模型假设,设计一种可调整上触碰点的触发型长寿债券,运用带永久跳跃的APC模型和双因素Wang转换定价方法对长寿债券进行定价,实证结果表明:在不同的参数组合下的风险溢价均处在一个合理的范围,由于模型参数多、可用死亡率数据年限短,风险溢价的结果对无风险利率等参数敏感性较高.     

A partial internal model for longevity risk

This paper proposes a simple partial internal model for longevity risk within the Solvency 2 framework. The model is closely linked to the mechanisms associated with the so-called Danish longevity benchmark, where the underlying mortality intensity and the trend is estimated yearly based on mortality experience from the Danish life and pension insurance sector, and on current data from the entire Danish population. Within this model, we derive an estimate for the 99.5% percentile for longevity risk, which differs from the longevity stress of 20% from the standard model. The new stress explicitly reflects the risk associated with unexpected changes in the underlying population mortality intensity on a one-year horizon and with a 99.5% confidence level. In addition, the model contains a component, which quantifies the unsystematic longevity risk associated with a given insurance portfolio. This last component depends on the size of the specific portfolio.     

Lee-Carter模型在中国城市人口死亡率预测中的应用与改进

由死亡率下降带来的长寿风险给社会、政治以及经济带来了新的挑战。为了更加准确地对长寿风险进行评估和管理,需要对未来死亡率趋势进行预测。本文针对我国死亡率数据样本量小以及数据存在缺失的实际情况,对Lee-Carter模型进行了改进,通过一个双随机过程对Lee-Carter模型中的时间项进行建模。在模型中考虑了样本量不足对预测结果造成的影响,使得改进后的Lee-Carter模型更加适合目前中国的人口死亡率预测。     

Cohort and value-based multi-country longevity risk management

ABSTRACT

Multi-country risk management of longevity risk provides new opportunities to hedge mortality and interest rate risks in guaranteed lifetime income streams. This requires consideration of both interest rate and mortality risks in multiple countries. For this purpose, we develop value-based longevity indexes for multiple cohorts in two different countries that take into account the major sources of risks impacting life insurance portfolios, mortality and interest rates. To construct the indexes we propose a cohort-based affine model for multi-country mortality and use an arbitrage-free multi-country Nelson–Siegel model for the dynamics of interest rates. Index-based longevity hedging strategies have the advantages of efficiency, liquidity and lower cost but introduce basis risk. Graphical risk metrics are a way to effectively capture the relationship between an insurer's portfolio and hedging strategies. We illustrate the effectiveness of using a value-based index for longevity risk management between two countries using graphical basis risk metrics. To show the impact of both interest rate and mortality risk we use Australia and the UK as domestic and foreign countries, and, to show the impact of mortality only, we use the male populations of the Netherlands and France with common interest rates and basis risk arising only from differences in mortality risks.     

A jump-diffusion approach to modelling vulnerable option pricing

Following the framework of Klein [1996. Journal of Banking and Finance 20, 1211–1229], this paper presents an improved method of pricing vulnerable options under jump diffusion assumptions about the underlying stock prices and firm values which are appropriate in many business situations. In contrast to Klein [1996. Journal of Banking and Finance 20, 1211–1229] model, jumps can be used to model sudden changes in stock prices and firm values. Further, with the jump risk, a firm can default instantaneously because of an unexpected drop in its value. Therefore, our model is able to provide sufficient conceptual insights about the economic mechanism of vulnerable option pricing. The numerical results show that a jump occurrence in firm values can increase the likelihood of default and reduce the vulnerable option prices.     

An Optimal Product Mix for Hedging Longevity Risk in Life Insurance Companies: The Immunization Theory Approach

This article investigates the natural hedging strategy to deal with longevity risks for life insurance companies. We propose an immunization model that incorporates a stochastic mortality dynamic to calculate the optimal life insurance–annuity product mix ratio to hedge against longevity risks. We model the dynamic of the changes in future mortality using the well‐known Lee–Carter model and discuss the model risk issue by comparing the results between the Lee–Carter and Cairns–Blake–Dowd models. On the basis of the mortality experience and insurance products in the United States, we demonstrate that the proposed model can lead to an optimal product mix and effectively reduce longevity risks for life insurance companies.     

两因子随机死亡率状态空间模型及长寿风险测度

引入状态空间模型对传统两因子CBD模型拟合阶段和预测阶段进行联合建模,并基于卡尔曼滤波方法对模型参数进行估计。进一步考虑到死亡率数据的小样本特征,结合Bootstrap仿真技术和生存年金组合折现模型对长寿风险进行测度。利用1996~2011年数据展开实证研究,结果表明:结合模型解释能力、参数估计结果和误差项正态分布检验结果,两因子状态空间模型要优于传统CBD模型;年金组合规模的扩大可以消除微观长寿风险,但不能消除宏观长寿风险和参数风险;宏观长寿风险占据着不可分散风险的主导地位。